Relationship between zeros and coefficients of a cubic polynomial Class 10 Maths | Polynomials

Understand it

This result extends the quadratic relationship to cubics.
It is useful when one or more zeros are known, or when a cubic polynomial is formed from given roots.
The signs must be read carefully from the standard form.
This relation saves time and is often used in higher-level algebra questions.

Example

For x³-6x²+11x-6, sum of zeros is 6, sum of pairwise products is 11, and product of zeros is 6.

Formula and rule

For zeros α, β, γ of ax³+bx²+cx+d: α+β+γ=-b/a, αβ+βγ+γα=c/a, and αβγ=-d/a.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

1For x³-3x²-4x+12, the product of zeros is -12/1=-12.
2If the zeros are 2, -2, and 3, their product is also -12.

Method you can reuse

1Compare the polynomial with ax³+bx²+cx+d.
2Note a, b, c, and d with signs.
3Apply the three formulas carefully.
4Use them to verify or construct the polynomial.

Visual tip

A cubic graph can cross the x-axis up to three times, and those intercepts correspond to the zeros used in these relations.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

What is the product of zeros of ax³+bx²+cx+d?

Think first, then show answer

The product of the zeros is -d/a. The negative sign is important, so students should read the constant term carefully from the standard form.

Common wrong answer

A common wrong answer is to write the product as d/a. The negative sign is required because the cubic relation changes sign for the constant term.

Don't mix this up

Students often forget that the product of three zeros has a negative sign with d/a in the standard form ax³+bx²+cx+d.

Exam tip

Write the cubic in standard form first, then apply all three relations one by one.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Find one relation from the coefficients, verify a set of roots, or construct a cubic from zeros.

1
Enough for 1 mark
If α, β, and γ are the zeros of ax³+bx²+cx+d, then α+β+γ=-b/a, αβ+βγ+γα=c/a, and αβγ=-d/a.
2
Add for 2 marks
Example: For x³-6x²+11x-6, sum of zeros is 6, sum of pairwise products is 11, and product of zeros is 6.
3
Add for 3 marks
This result extends the quadratic relationship to cubics.

Use the concept graph around this idea instead of revising it in isolation.

Cubic polynomial and its zeros

A cubic polynomial is a polynomial of degree 3, usually written as ax³+bx²+cx+d where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.

Forming cubic polynomial from given zeros

If α, β, and γ are the zeros of a cubic polynomial, then the polynomial can be written as k(x-α)(x-β)(x-γ), where k is a non-zero constant.

Verification of given zeros

To verify whether a number is a zero of a polynomial, substitute that number into the polynomial and check whether the value becomes 0.

Geometrical meaning of zero of a polynomial

A zero of a polynomial is the value of x for which the polynomial becomes 0. On the graph, it is the x-coordinate where the curve touches or cuts the x-axis.

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Relationship between zeros and coefficients of a cubic polynomial

Understand it

See it worked out

Check yourself

Write it in the exam

Practice this concept with focused MCQs

Help improve this page

Relationship between zeros and coefficients of a cubic polynomial

Understand it

See it worked out

Check yourself

Write it in the exam

Related concepts

Practice this concept with focused MCQs

Help improve this page